White House Considers Using Storm Aid Funds as a Way to Pay for the Border Wall

“I would like to do a much broader form of immigration,” Mr. Trump said. “We could help the Dreamers.”

Only hours earlier, Mr. Pence had rejected such a deal, saying the president wanted to wait until the Supreme Court ruled this spring on whether the program, Deferred Action for Childhood Arrivals, or DACA, was constitutional. “No wall, no deal,” Mr. Pence declared in a briefing with reporters on Capitol Hill. “We’re going to keep standing strong, keep standing firm.”

The vice president faulted the Democrats, but he has essentially blocked potential solutions for the impasse. He has made it clear that Mr. Trump would not drop his insistence on funding for a wall on the southwestern border, which Democrats have branded a nonstarter.

Mr. Pence also indicated that the president was disinclined to accept the idea behind a bipartisan plan that had been under discussion in the Senate that would trade wall funding for legal status for undocumented immigrants facing the threat of deportation, including the Dreamers and people who previously held Temporary Protected Status.

Privately, he told Mr. Graham’s group that the president also would not support a proposal that would reopen the government for three weeks while Republicans and Democrats work to hash out a broader legislative deal on the wall and temporary grants of legal status for the two groups.

“We’re kind of stuck,” Mr. Graham conceded.

Speaker Nancy Pelosi, Democrat of California, also showed no signs of budging, urging the Republican-controlled Senate to take up a measure that the House passed on Wednesday to reopen part of the government. The House passed two more measures on Thursday, this time funding the Departments of Housing and Urban Development, Transportation and Agriculture, as well as the Food and Drug Administration.

A dozen Republicans crossed party lines to support one of the measures — slightly more than in previous votes, but no indicator that the patience of Mr. Trump’s own party was wearing thin.